A-Level Mathematics May/June 2025 Q11(a): The function f is defined by f(x) = x² + 4ax + a for x ∈ R, where a is a constant. The…

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  The function is given by $f(x) = x^2 + 4ax + a$. The range of the function is given as $f(x) \ge -33$. This means the minimum value of the quadratic function is $-33$.

  To find the minimum value, we complete the square for $f(x)$. $f(x) = (x + 2a)^2 - (2a)^2 + a$ $f(x) = (x + 2a)^2 - 4a^2 + a$

  The vertex of the parabola is at $(-2a, -4a^2 + a)$. The minimum value of the function is the y-coordinate of the vertex. Minimum value = $-4a^2 + a$.

  We are given that the minimum value is $-33$. Therefore, we can set up an equation: $-4a^2 + a = -33$

  Rearranging this into a standard quadratic form ($Ax^2+Bx+C=0$): $0 = 4a^2 - a - 33$

  We can solve this quadratic equation for $a$ by factorisation. $(4a + 11)(a - 3) = 0$

  This gives two possible values for $a$: $4a + 11 = 0 \implies 4a = -11 \implies a = -\frac{11}{4}$ $a - 3 = 0 \implies a = 3$

  The possible values of $a$ are $3$ and $-\frac{11}{4}$.

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  How the marks are awarded

  • B1 — Correctly completing the square to get the expression $(x+2a)^2 - 4a^2 + a$.
  • M1 — Equating their expression for the minimum value, $-4a^2+a$, to the given value of $-33$.
  • B1 — Obtaining the correct quadratic equation $4a^2 - a - 33 = 0$ or an equivalent form.
  • A1 — Finding both correct values for $a$, which are $3$ and $-\frac{11}{4}$ (or $-2.75$).

  Common mistakes

  • Errors in completing the square, for example writing $(x+2a)^2+a$ and forgetting to subtract the $(2a)^2$ term.
  • Sign errors when identifying the minimum value from the completed square form, for instance stating the minimum is $4a^2+a$ instead of $-4a^2+a$.
  • Mistakes when solving the quadratic equation $4a^2 - a - 33 = 0$, often through incorrect factorisation or sign errors in the quadratic formula, leading to incorrect values for 'a'.
  • Incorrectly rejecting one of the solutions, for example discarding $a = -11/4$ under the false assumption that 'a' must be a positive value or an integer.

  Examiner tip: To find the minimum or maximum value of any quadratic function, the most reliable method is completing the square to identify the coordinates of the vertex.

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